On Mean Convergence of Lagrange Interpolation for General Arrays |
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Authors: | D S Lubinsky |
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Institution: | Department of Mathematics, Centre for Applicable Analysis and Number Theory, Witwatersrand University, Wits, 2050, South Africaf1 |
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Abstract: | For n1, let {xjn}nj=1 be n distinct points in a compact set K
and letLn·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {xjn}1jn, n1 ensure the existence of p>0 such that limn→∞ (f−Lnf]) vLp(K)=0 for very continuous f: K→
? We show that it is necessary and sufficient that there exists r>0 with supn1 πnvLr(K) ∑nj=1 (1/|π′n| (xjn))<∞. Here for n1, πn is a polynomial of degree n having {xjn}nj=1 as zeros. The necessity of this condition is due to Ying Guang Shi. |
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